`solver_abtb`

mixed systems solver

# Synopsis

        solver_abtb stokes     (a,b,mp);
        solver_abtb elasticity (a,b,c,mp);

# Description

This class provides both direct and iterative algorithms for solving mixed problem:

       [ A  B^T ] [ u ]    [ Mf ]
       [        ] [   ]  = [    ]
       [ B  -C  ] [ p ]    [ Mg ]

where A is symmetric positive definite and C is symmetric positive. By default, iterative algorithms are considered for tridimensional problems and direct methods otherwise. A `solver_option` argument can change this default behavior. Mixed linear problems appears for instance with the discretization of Stokes and elasticity problems. The C matrix can be zero and then the corresponding argument can be omitted when invoking the constructor. Non-zero C matrix appears for of Stokes problems with stabilized P1-P1 element, or for nearly incompressible elasticity problems.

Recall that, for 1D and 2D problems, the direct method is default, since it is more efficient: see e.g. User's guide. The `solver_option` allows one to change this default behavior.

# Direct algorithm

When the kernel of `B^T` is reduced to zero, the global system is non-singular and the `solver` method could be applied. Otherwise, when the kernel of `B^T` is not reduced to zero, then the pressure p is defined up to a constant and the system is singular. This is a major difficulty for any direct method. Thus, the system is first completed by the imposition of an additional constraint on the pressure term: it should have a zero average value. By this way, the system becomes non-singular and the `solver` method could be applied.

# Iterative algorithm

The `cg` preconditionned conjugate gradient algorithm or the `gmres` one is used, depending on the symmetry of the matrix. The `mp` matrix is used as preconditionner: it can be customized by the `set_preconditionner` member function. The linear sub-systems related to the `A` matrix are also solved by an inner solver. This inner solver is automatically defined by default: it can be customized by the `set_inner_solver` member function and e.g. uses it own inner iterative algorithm and preconditionner.

# Example

See the User's guide for practical examples for the nearly incompressible elasticity, the Stokes and the Navier-Stokes problems.

# Implementation

This documentation has been generated from file linalg/lib/solver_abtb.h